Problem: You have found the following ages (in years) of all 5 sloths at your local zoo: $ 8,\enspace 11,\enspace 13,\enspace 11,\enspace 11$ What is the average age of the sloths at your zoo? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we have data for all 5 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{8 + 11 + 13 + 11 + 11}{{5}} = {10.8\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $8$ years $-2.8$ years $7.84$ years $^2$ $11$ years $0.2$ years $0.04$ years $^2$ $13$ years $2.2$ years $4.84$ years $^2$ $11$ years $0.2$ years $0.04$ years $^2$ $11$ years $0.2$ years $0.04$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{7.84} + {0.04} + {4.84} + {0.04} + {0.04}} {{5}} $ $ {\sigma^2} = \dfrac{{12.8}}{{5}} = {2.56\text{ years}^2} $ The average sloth at the zoo is 10.8 years old. The population variance is 2.56 years $^2$.